The key idea for the further extension of \(\partial _t^{\alpha }\) ,
is the family \(\lbrace  H_{\alpha }(0,T)\rbrace _{\alpha > 0}\)  and
the isomorphism \(\partial _t^{\alpha }: H^{\alpha +\beta }(0,T) \,\longrightarrow \, H^{\beta }(0,T)\)  with \(\alpha ,\beta >0\) , which is called
a Hilbert scale. We can refer to Chapter V in Amann [1]} as for
a general reference.
